Abstract

Introduction

A non-cosmologist, reading a statement in the press, might find something like this: "A very distant galaxy has just been discovered. It is so far away that the light from it that we see now left the galaxy when the universe was less than 10% of its present age and has been traveling for 14 billion years. The galaxy is 14 billion light years away. The redshift of its light is 5.8." This statement seems confusing. If the galaxy is now 14 billion light years away, it would seem to have been only about 1 billion light years away or so when the light left it. Why did it take 14 billion years for light to travel 1 billion light years? On the other hand if the galaxy was 14 billion light years away when the light left it 1 billion years after the big bang,, was it going at 14 times the speed of light or more? The equations for the relativistic Doppler shift of light would give a velocity of about 96% of the speed of light for this redshift. This set of statements would seem to be puzzling.

In order to understand what is going on, we need to look at concepts of general relativity and to consider the various models cosmologists have developed. We need to know the results of various observations and measurements of cosmological quantities. We need to understand that the popular press may at times oversimplify the descriptions of cosmological findings. There are exciting new developments in cosmology and new tools will bring in a flood of new data in the next few years.

Edwin Hubble made many measurements and found that the velocity of recession, v, as determined from the redshift, was approximately proportional to the distance from us, d, determined from the apparent luminosity. The velocities he measured were small compared to the velocity of light, c. This observation created the Hubble constant, H, as in the equation v=Hd. Later measurements extended the range of this relationship but still supported the form v=Hd. This simple equation suggests that if we run time backwards and assume a constant velocity for each galaxy, there was a point in time when all the galaxies in a particular finite region of space such as the presently visible universe were closer together in the past and the universe was more dense.

If we think of this contraction as occurring at a constant rate then there was a time in the past when the space represented by that presently visible universe was extremely dense and, in fact, reached a condition of such extreme matter-energy density and temperature and pressure that the present laws of physics no longer held. If we arbitrarily continue the extrapolation backward to a point of apparent infinite density, we can take this time as t=0 and think of the rapid expansion of space itself after that time as the "big bang". The time after the big bang, the present time, can be called t_o. This time is simply 1/H in suitable units for the case where the velocity of expansion has been taken to be constant during this period.

Cosmologists generally think in terms of an extremely short time like 10^-43 seconds for the initial period when the density was too high for present physics to hold followed by a very rapid expansion period called inflation. All of this taking place in something like 10^-32 seconds. Following this, came a period of a very dense, rapidly expanding universe dominated by radiation rather than matter. This period lasted some 300,000 years or so. After this period radiation and matter decoupled and the radiation left over from that period now stretched in wavelength by something like a factor of 1000 is the cosmic microwave background radiation (CMBR) of such interest to cosmologists today. At about the same epoch the universe became matter dominated and we can neglect radiation density compared to matter density for significantly later periods.

The present age of the universe is generally taken to be something like 15 billion years. 300,000 years is thus 0.002% of this total age. Although these early periods are of great interest to cosmologists we will consider in this paper only models that apply after this phase or for more than 99.99% of the history of the universe as we are interested in questions of the present and future rate of expansion and such things as lookback times and distances of distant galaxies. It should be noted in this picture however, that if our present universe is finite in extent it would have been squeezed to a minute speck at the beginning, but if the universe now is infinite in extent it could be infinitely dense and still infinite in extent at the beginning even though any particular finite volume such as the presently visible universe would be just a speck.

While H may be a constant at a given time over galaxies at many distances, it is not a constant over time itself during the period of interest in this paper for the commonly used models considered. Since H=v/d, if we track a given galaxy as space expands (and neglect its own proper motion which is a good approximation except for close-in galaxies) we will see that as time goes on for those models in which the recession velocity, v, is a constant or decreases with time and since d is increasing with time for an expanding universe, H will decrease with time. A recent set of measurements of distant supernovae indicates that perhaps the expansion rate is now actually increasing somewhat but this set of measurements still indicates insufficient increase with v in time to offset the increase in d as galaxies move away, hence H is decreasing at the present time in this case also.

For several models of this paper H in fact varies as 1/t. For the last model, Model V, H approaches a constant value eventually. The present value of H is called H_o. Recent measurements suggest a value in the units usually used of between 50 and 80 kilometers per second per megaparsec for this constant ( a parsec is 3.26 light years). This would give a value for t_o of 19.6 to 12.2 billion years if galaxies have receded at constant velocities during this time. This is called the Hubble time or 1/H_o in suitable units. More about this later as specific models are discussed.

A cosmological model as discussed in this paper is a mathematical model used by cosmologists to relate, among other things, redshift and luminosity to distances and look-back times and velocities of recession and to realize that there might be two distances, the distance a galaxy is away now as we receive its light, D_r, and the distance away it was when the light was emitted that we see now, D_e. There are also two times to be considered, the time since the big bang when the light was emitted that we see now, t_e, and the time when the light is received, t_r (which is also called t_o when it is the present time). There are several different cosmological models in use by cosmologists and they give somewhat different results for these distances and times for any given observation of luminosity and redshift. They also give somewhat different results for the future development of the universe. New observational results continue to refine and select the models and suggest new ones.

There will be a redshift in the spectral lines of each galaxy related to its recession velocity. For this model, this redshift is called the Doppler shift. The redshift, z, is defined as z=(l-l_o)/l_o so that z+1=l/l_o which is simply the observed wavelength at our telescope of a particular spectral line divided by the emitted wavelength (the wavelength measured in the laboratory). For low velocities and hence low values of z, this leads to the relation v=cz. For velocities of recession approaching that of light we need to use the equations of special relativity in this model to get the relativistic Doppler effect equation tying the recession velocity, v, to the redshift, z. The equations are:

v=c((z+1)²-1)/((z+1)²+1)--------(1)
z+1=((1+v/c)/(1-v/c))^1/2--------(1a)

There is a problem with this model. Consider a galaxy rushing away from us at a constant velocity of very nearly the speed of light. Light that it emitted when the universe was half as old as it is now would just be getting to us now and we could not have a lookback time, t_lb, (t_lb=t_o-t_e), greater than half the age of the universe. We find we are able to see much further back in time than this. The cosmic microwave background radiation, CMBR, coming to us from all directions was emitted when the universe was very young (something like 300,000 years after the big bang or about 1/50,000 of its present age), and we can detect this radiation now. It is this background radiation that greatly strengthened the arguments for the big bang theory. We are also able to look back now to galaxies as they were when the universe was less than 10% of its present age. Also, consider a light packet just reaching us now. In this first simple model that packet coming toward us, if it started at nearly the birth of the universe, would have been some 15 billion light years away from us at the big bang - but the big bang model assumes that all of the presently observable region of space was in a very tiny volume at that time.

The delay and deflection of light signals passing close to massive objects have now been measured with increasing precision and are in agreement with the predictions of general relativity at the 0.1% level. Geodetic precession has been detected by the laser ranging to the moon coupled with radiointerferometry data. Gravitational radiation from accelerated masses in a binary pulsar system has been shown to be consistent with general relativity at the 0.4% level.

There are still untested aspects of general relativity and there are still competing theories. It is planned to fly a NASA experiment called Gravity Probe B to test general relativity predictions for an effect called frame dragging (to 1%) and geodetic effects (to a part in 10,000) and to see how other theories fare. (The launch date continues to slip). The concepts and equations of general relativity are used in the development of most cosmological models favored by cosmologists, however, and these concepts are used in what follows.

One such concept of general relativity is that we should not think of galaxies rushing away from us through a fixed space, we should think of galaxies more or less standing still in their local space but that space itself is expanding. This is a difficult concept but it can be described mathematically. In this concept, light leaving a distant galaxy is traveling at velocity c toward us in that local space but since space itself is expanding and that local space is moving away from us, the velocity of a light packet toward us (defined in a certain way) is less than c until the packet gets to our local space and the velocity can even be negative for some early portion of the time light travels to us.

Cosmologists also use a principle called the Cosmological Principle which says that if we look at sufficiently large regions of space, space is homogeneous and isotropic, the same everywhere and the same in all directions, there is no center. This means that the rate of expansion of space can be expressed in terms of a single scale factor R(t) which is the same at all points of space at any one time but will be a function of time for an expanding or contracting universe.

The proper distance (explained below) between any two galaxies as a function of time is proportional to R(t) (neglecting small local motions of the galaxies as we will do in most of what follows).

Another concept has to do with the apparent luminosity of distant galaxies. If we define the absolute luminosity, L_A, as the total radiant power from a galaxy over all angles and define the apparent luminosity, L_a, as the radiant power reaching us per unit area of our telescope, then in the simple case of a nearby galaxy, L_a=L_A/4pd² where d is the distance to that galaxy, hence d=(L_A/4pL_a)^1/2. This is called the luminosity distance, D_l. (There may have to be an additional correction, called the K correction, for the shape of the energy spectrum of the emitting galaxy and the limitations of the spectral range of the measuring instruments which measure L_a.) (L_A is based on observations of nearby galaxies and assumes a fairly consistent total radiant power of even the distant galaxies of similar type.)

Since for more distant galaxies the light has been traveling toward us for a long time, the question arises, is this the proper distance away the galaxy was when it emitted the light we see now, or the proper distance away it is now or even something else? With this concept of general relativity of space itself expanding, reflection will show that the correct distance to use is the proper distance away now when the light is received, D_r.

There is also a need to make a correction for the wavelength of the received photons as the light has been stretched out as the universe has expanded during its travel. This reduces the energy by the factor of 1/(z+1). Also, the rate at which the photons are being received is reduced by the same factor so L_a is reduced by 1/(z+1)². The final equation then for D_r is:

This distance is called the proper distance or proper comoving distance. See Narlikar, Introduction to Cosmology pp 94-97⁽¹⁾ for this derivation.

Now we have several concepts of distance. To understand the proper distance let's use a simple analogy. Suppose our universe is only two-dimensional and is modeled by the surface of a balloon which is being steadily inflated. A sticker on the balloon represents our position and other stickers represent the positions of other galaxies. At a particular instant let's measure with a tape along the surface of the balloon the distance from our sticker to another sticker representing the galaxy under observation. This measurement is the proper distance. Another way to describe it is to imagine starting at our position in space in the real universe with a series of observers in a straight line to the galaxy being examined, each carrying a non-expanding measuring stick and measuring the distance to his neighbor at a particular instant. The sum of these measurements is the proper distance at that instant. This seems like the most physically meaningful distance concept. So now, we have the luminosity distance, D_l, the proper distance now, D_r, and the proper distance then, D_e.

There is one more concept of distance, the light travel distance. Imagine an insect crawling along the surface of the balloon from the distant sticker to our sticker. If the insect counted its number of steps to get here and knew the length of each step and multiplied these numbers, it would come up with a distance greater than the distance then but less than the distance now. Call this distance D_lt or the light-travel distance. Thus D_e<D_lt<D_r. Thus, D_lt=ct_lb. Note that we now have four concepts of distance. The differences are illustrated in a later Figure related to the third model to be discussed.

One can also think of the analogy of a muffin filled with raisins. Before the muffin is baked, the raisins are close together. As it is baked and the yeast causes it to rise, the raisins all move away from each other, and the further apart two raisins are at the beginning the more rapidly they move away from each other. The distance apart of two raisins is the proper distance in this analogy when this separation is measured with a ruler.

The balloon analogy is helpful in another way. The balloon surface is a closed 2-dimensional curved surface of finite area, yet it has no edges. One can imagine it starting from almost a point then inflating to a certain size, stopping and contracting again. Another surface would be a flat surface of infinite extent. Still another would be something like a saddle-shaped surface, also of infinite extent.

These illustrate in 2 dimensions what some of the concepts of general relativity are of curved 3-dimensional surfaces which we cannot readily visualize but which can be described mathematically. The balloon analogy carried to a higher dimension is called a hypersphere.

For the cosmological models most used by cosmologists, (neglecting something called the cosmological constant which will be explained later), if the present mass density of the universe is large enough, space is positively curved (like the balloon). If the mass density is a certain critical value, space is flat. For lower mass densities, space is negatively curved, something like saddle-shaped. (In a space of positive curvature the sum of the interior angles of a triangle will be greater than 180⁰, for flat space, the sum of the angles will equal 180⁰, and for negatively curved space the sum will be less than 180⁰. You would need an enormously large triangle to see this effect.) For a positively curved (or closed) universe, gravity is strong enough that the expansion will slow down, stop and reverse. The volume of this universe will be finite, and so will its lifetime. For the universe of just the critical density, the expansion will slow down and only stop after an infinite time (dR/dt-->0). Such a universe is generally considered to be open or infinite in extent and to be flat. At lesser densities the universe (also open) will expand forever. For a universe of this present lower mass density, if we look back in time the density approaches the critical density closer and closer to be virtually identical to it just after the big bang. For a description of these spaces see The Shape of Space⁽²⁾ by Jeffrey Weeks.

The balloon analogy helps us to think about questions like, if space is flat or saddle-shaped, does space have an edge? Is there more of the universe we cannot now see? Are there more universes different from ours? These are difficult questions but the balloon analogy helps us to see that for at least some universe models space can be finite in extent but still not have an edge. There is a recent article in the Scientific American⁽³⁾ discussing these questions of the extent and shape of space and suggesting possibilities for space to be flat or saddle-shaped but still finite in what is referred to as a multiply-connected universe. In the next few years, more observations of galaxy distributions may shed light on this issue. Some cosmologists think that model universes which have zero or negative curvature and yet are finite in extent are "pathological" while others want to search for evidence that our universe might be of such a kind.

Another concept of general relativity is that the red shift of light from distant galaxies due solely to the expansion of space itself is not given by the relativistic Doppler shift equation of special relativity but is simply related to the present scale factor of the universe compared to the scale factor when the light was emitted that we see now. The equation is:

The relativistic Doppler effect is appropriate for the redshifts due to local motions such as the rotation of our own Milky Way galaxy and its motion toward a nearby galaxy cluster, the Virgo cluster.

The equation for the total redshift when both the cosmological redshift and Doppler redshift are involved is given in Peebles Principles of Physical Cosmology(4), pages 96-98.

While in special relativity there is no standard of rest and no preferred reference frame, in general relativity the situation is somewhat different. A rest frame for any particular region of space can be defined as the frame which is not rotating compared to the background of distant stars and for which the cosmic microwave background radiation, CMBR, has the same spectrum in all directions and so the concept of local motion in that frame is meaningful. In another region of space, there would also be a rest frame for which the CMBR is the same in all directions, but the two rest frames are moving with respect to each other so there is no preferred frame of that type for the universe as a whole and no center of the universe. We can however, define a comoving rest frame for the universe as a whole for which the requirement of the CMBR being the same in all directions and there being a standard of rotation based on the distant stars is met at all locations.

Still another concept is that, since we are thinking of space itself expanding rather than of galaxies rushing through space, and since we are assuming that galaxies are nearly standing still in their local spaces for these simple models, we need not use the equations of special relativity of the difference in clock rates of observers in uniform motion with respect to each other, we can think, instead, of a single universal time for the universe as a whole.

General relativity also allows for the introduction of something called the cosmological constant. For now, assume that the cosmological constant is not involved in the first few models to be discussed.

There is evidence of "dynamic dark matter" in halos around galaxies deduced from the pattern of the orbital motions of the stars and gas in these galaxies. Just recently, some of this "dark matter" has been seen faintly in the infrared halo of a nearby galaxy and information is beginning to come in from the X-ray satellite. Some dark matter also exists between galaxies in galaxy clusters. Some of this dark matter might be dead or dying stars. There is even some evidence that neutrinos might have a very small mass and further add to the total, and there may be other forms of dark matter of even more exotic nature called Machos and Wimps, but it is hard to find evidence for more than about 1/3 the critical mass from all known sources.

For some years the cosmological constant has not been used much or has been set equal to zero in most models and has been set equal to zero in the discussion thus far. Recently, however, there has been great renewed interest in a non-zero cosmological constant. If we use it in a model, and still start with the big bang and choose a certain range of values for the cosmological constant, we will get a universe which expands very rapidly at first, then slows down like our Model III but stops decelerating at some point and then begins to accelerate for ever after. Space can be flat in such a universe.

Note that the distance is not ct₀ as many books and journal articles suggest but 3ct₀. (Maybe they are using the light travel distance or not grasping these concepts.) This result clearly implies that a galaxy can be receding from us at a proper velocity due to the expansion of space itself greater than the speed of light and still be seen by us since the redshift is not given by the relativistic Doppler shift equation but by equation (3) for a galaxy at rest in its own space.

Another concept is called the event horizon, the minimum distance away a particle can be now so that the light it emits now will not reach our location in any finite time in the future. For Model III there is no event horizon, the light emitted now by any particle in the universe will eventually reach our location.

• What cosmological model is being used?
• Is space flat or curved positively or negatively?
• What concept of distance is being used?
• If proper distance, is it distance now or distance then?
• What value of H₀ is being used?
• Is the redshift treated as cosmological or Doppler?
• What value of the cosmological constant is being used?
• What is the assumed ratio of actual mass density to critical density?